Tuesday, April 16, 2019


Filed under: General — m759 @ 11:00 AM

Related material — Posts tagged Perspective.

Saturday, November 3, 2018

Foundations of Geometry

Filed under: G-Notes,General,Geometry — m759 @ 1:40 PM

"costruire (o, dirò meglio immaginare) un ente" — Fano, 1892

"o, dirò meglio, costruire" — Cullinane, 2018

Saturday, July 1, 2017


Filed under: General — m759 @ 4:40 PM

Remarks by the director of "Inception" —

Related material —

Thursday, June 4, 2015


Filed under: General,Geometry — Tags: — m759 @ 7:47 PM

Princeton University's president on June 2, 2015 —

“Dream Audaciously,” Eisgruber Urges Graduates

Related news —

Film director Christopher Nolan at Princeton on June 1, 2015:

“In these graduation speeches, generally, you have the
speaker say something along the lines of, ‘You need to
chase your dreams,’ ” Nolan said. “But I’m not going to
say that because I don’t believe it. I don’t want you to
chase your dreams. I want you to chase your realities.
And I want to say: Don’t chase your realities at the
expense of your dreams, but as the foundation of your

IMAGE- Right 3-4-5 triangle with squares on sides and hypotenuse as base

"If you have built castles in the air, 
your work need not be lost;
that is where they should be.
Now put the foundations under them.”

— Henry David Thoreau

Sunday, May 31, 2015


Filed under: General,Geometry — m759 @ 6:00 AM

IMAGE- Desargues's theorem in light of Galois geometry

To enlarge image, click here.

Saturday, October 25, 2014

Foundation Square

Filed under: General,Geometry — Tags: — m759 @ 2:56 PM

In the above illustration of the 3-4-5 Pythagorean triangle,
the grids on each side may be regarded as figures of
Euclidean  geometry or of Galois  geometry.

In Euclidean geometry, these grids illustrate a property of
the inner triangle.

In elementary Galois geometry, ignoring the connection with
the inner triangle, the grids may be regarded instead as
illustrating vector spaces over finite (i.e., Galois) fields.
Previous posts in this journal have dealt with properties of
the 3×3 and 4×4 grids.  This suggests a look at properties of
the next larger grid, the 5×5 array, viewed as a picture of the
two-dimensional vector space (or affine plane) over the finite
Galois field GF(5) (also known as ℤ5).

The 5×5 array may be coordinatized in a natural way, as illustrated
in (for instance) Matters Mathematical , by I.N. Herstein and
Irving Kaplansky, 2nd ed., Chelsea Publishing, 1978, p. 171:

See Herstein and Kaplansky for the elementary Galois geometry of
the 5×5 array.

For 5×5 geometry that is not so elementary, see…

Hafner's abstract:

We describe the Hoffman-Singleton graph geometrically, showing that
it is closely related to the incidence graph of the affine plane over ℤ5.
This allows us to construct all automorphisms of the graph.

The remarks of Brouwer on graphs connect the 5×5-related geometry discussed
by Hafner with the 4×4 geometry related to the Steiner system S(5,8,24).
(See the Miracle Octad Generator of R. T. Curtis and the related coordinatization
by Cullinane of the 4×4 array as a four-dimensional vector space over GF(2).)

Saturday, July 20, 2013


Filed under: General — m759 @ 11:45 PM

"Dichtung ist Stiftung." — Heidegger

Monday, October 1, 2012


Filed under: General,Geometry — m759 @ 3:09 PM

IMAGE- 'Square Acquires a New York Design Firm' (NY Times, noon on October 1st, 2012)

» more

This, together with this morning's post involving the squares 16 and 9,
suggests a review of Conceit at Harvard (October 25, 2006), which
contains the following figure involving the squares 16, 9, and 25—

IMAGE- Right 3-4-5 triangle with squares on sides and hypotenuse as base

"If you have built castles in the air, 
your work need not be lost;
that is where they should be.
Now put the foundations under them.”

— Henry David Thoreau

Saturday, January 11, 2020

Mathematics or Theology?

Filed under: General — m759 @ 10:12 AM

Hersh wrote a paper with a title containing the phrase 
“The Kingdom of Math is Within You.”

In his memory, see Log24 posts from the date of his death
tagged Inner-Space Variations.

Related literature:  Hersh's "Death and Mathematics Poems."

See as well this  journal on the above publication date.

Wednesday, November 6, 2019

Soul Snatchers

Filed under: General — m759 @ 11:18 PM

From All Souls' Day 2015

George Boole in image posted on All Souls' Day 2015

Related entertainment —

Invasion of the Soul Snatchers (Wild Palms  review, 1993).

Sunday, September 15, 2019

For Weisheit, the Rabbi

Filed under: General — m759 @ 3:30 AM

See also Log24 remarks from the date of Greene's death
in posts now tagged Bregnans

A page related to Eleanor Cook's "wonderful first line"
quoted above from A Reader's Guide to Wallace Stevens :

A Reader's Guide to Eleanor Cook

Monday, September 9, 2019

The Montenegro Contingency

Filed under: General — m759 @ 8:44 AM

(For Harlan Kane)

"While digging in the grounds for the new foundation,
the broken fragments of a marble statue were unearthed."

— From Thomas Hardy, "Barbara of the House of Grebe,"
quoted in an epigraph to Paul de Man's "Shelley Disfigured,"
in turn quoted by Barbara Johnson on page 231 of Persons
and Things  
(Harvard paperback, 2010).

From "the world of the unintentional, the contingent, the minute,
and the particular" (Kovacevic, U. of Montenegro, 2011) —

Yes, we received your payment.
No, it wasn't late, but it was for $78.13,
and the bill was for $78.31.
Okay, great.

Read more: https://www.springfieldspringfield.co.uk/

Another such transposition:  Pages  213 and 231 in a search
for "gaps" in a 2010 paperback discussion of Lacan —

These pages are as follows —

Tuesday, August 20, 2019

Ice Giants and Fire Gods: Mind the Gap

Filed under: General — Tags: — m759 @ 5:52 PM

From my reading Monday morning —

From the online New York Times  this afternoon —

Related literature —

For the Church of Synchronology

The Gigantomachia page above is dated September 20, 2003.
See as well my own webpage from that date: "The Form, the Pattern."

Monday, July 1, 2019

“The Ontological Secret”

Filed under: General — Tags: — m759 @ 3:57 AM

The phrase "ontological secret" is from 1927 —

" Beauty is thus 'a flashing of intelligence…
on a matter intelligibly arranged' or, as Maritain
adds in the 1927 edition of Art and Scholasticism ,
it is 'the ontological secret that [things] bear within
them[selves], their spiritual being, their operating
mystery.' "

— John G. Trapani, Jr., "'Radiance': The Metaphysical Foundations
of Maritain's Aesthetics," pp. 11-19 in Beauty, Art, and the Polis ,
ed. by Alice Ramos, publ. by American Maritain Association, 2000.

This 1927 phrase may be the source of McLuhan's 1944
"ontological secret" —

From a search in this  journal for "Object of Beauty" —

“She never looked up while her mind rotated the facts,
trying to see them from all sides, trying to piece them
together into theory. All she could think was that she
was flunking an IQ test.”

— Steve Martin, An Object of Beauty

Sunday, May 19, 2019

The Building Blocks of Geometry

Filed under: General — Tags: , — m759 @ 10:21 PM

From "On the life and scientific work of Gino Fano
by Alberto Collino, Alberto Conte, and Alessandro Verra,
ICCM Notices , July 2014, Vol. 2 No. 1, pp. 43-57 —

" Indeed, about the Italian debate on foundations of Geometry, it is not rare to read comments in the same spirit of the following one, due to Jeremy Gray13. He is essentially reporting Hans Freudenthal’s point of view:

' When the distinguished mathematician and historian of mathematics Hans Freudenthal analysed Hilbert’s  Grundlagen he argued that the link between reality and geometry appears to be severed for the first time in Hilbert’s work. However, he discovered that Hilbert had been preceded by the Italian mathematician Gino Fano in 1892. . . .' "

13 J. Gray, "The Foundations of Projective Geometry in Italy," Chapter 24 (pp. 269–279) in his book Worlds Out of Nothing , Springer (2010).

Restoring the severed link —

Structure of the eightfold cube

See also Espacement  and The Thing and I.

Related material —


Tuesday, May 14, 2019


Filed under: General — Tags: — m759 @ 10:02 PM

"Is there no change of death in paradise?" — Wallace Stevens 

From a New Yorker  book review dated May 13 —

"In 'Field Flowers,' Glück’s flower scoffs that 'absence of change' 
is humanity’s 'poor idea of heaven.' But the religious believer
might object that Hägglund’s idea of eternity is equally poor." 

Here James Wood is reviewing Martin Hägglund’s
This Life: Secular Faith and Spiritual Freedom 
(Pantheon, March 5, 2019).

See also posts tagged Change Arises in this  journal.

Hägglund himself appeared here on June 21, 2014.

Tuesday, April 23, 2019


Filed under: General — m759 @ 12:53 PM

A check of an author cited in the previous post yields —

The cover illustration above is by Kay Nielsen . . . 


(Paris Review , December 3, 2015.)

See as well the April 1977 poem "Winter Tree" by Jon Lang and . . .

Sunday, March 17, 2019

Just Another Block in the Wall

Filed under: General — m759 @ 3:01 PM

Yesterday's post Grundlagen  —

Midrash on yesterday's Grundlagen

A poem linked to here on the above "building blocks" date, in the
Log24 post Sermon of  11 AM ET Sunday, 15 September 2013 —

Wednesday, January 16, 2019

Permutahedron Dream

Filed under: General — Tags: , , , — m759 @ 3:21 PM

The geometric object of the title appears in a post mentioning Bourgain 
in this journal.  Bourgain appears also in today's online New York Times —


Bourgain reportedly died on December 22.

An image from this journal on that date

Related poetic meditations —

IMAGE- Herbert John Ryser, 'Combinatorial Mathematics' (1963), page 1

Wednesday, January 9, 2019

Desperately Seeking Clarity*

Filed under: General — Tags: , , — m759 @ 2:10 PM

A much earlier, much truer, obituary —

* A sequel to Resonant Clarity and Desperately Seeking Resonance.

  Quantum electrodynamics  is also known as QED.

Monday, December 31, 2018

Eve’s Riddle

Filed under: General — m759 @ 12:00 PM

The Dickinson poem quoted above is numbered 373 at 
the Poetry Foundation.

See also Eternity + 373 in this  journal.

Saturday, December 29, 2018

Winter Tree

Filed under: General — m759 @ 1:05 PM

The title is that of a 1977 poem by Jon Lang.

A different sort of narrative tree, from The Onion  on the
date Friday, July 1, 2016, was illustrated in the previous post.

Material from this  journal related to July 1, 2016,
and the following day, a Saturday, is now tagged
"Saturday Night in the Labyrinth."

"A Damned Serious Business." — Rex Harrison on comedy

Friday, December 28, 2018

Phenomenology of Viewing

Filed under: General — Tags: — m759 @ 11:58 PM

From a post of December 22, 2018

See as well related posts now tagged Blackline.

Saturday, December 22, 2018

The Hat Tip

Filed under: General — Tags: , — m759 @ 9:59 PM

"Form the turtle!"
— Rex Harrison, quoted here
on March 12, 2005.

Thursday, November 8, 2018

Reality vs. Axiomatic Thinking

Filed under: G-Notes,General,Geometry — m759 @ 11:16 PM

From https://blogs.scientificamerican.com/…

A  Few  of  My  Favorite  Spaces:
The Fano Plane

The intuition-challenging Fano plane may be
the smallest interesting configuration
of points and lines.

By Evelyn Lamb on October 24, 2015

"…finite projective planes seem like
a triumph of purely axiomatic thinking
over any hint of reality. . . ."

For Fano's axiomatic  approach, see the Nov. 3 Log24 post
"Foundations of Geometry."

For the Fano plane's basis in reality , see the eightfold cube
at finitegeometry.org/sc/ and in this journal.

See as well "Two Views of Finite Space" (in this journal on the date 
of Lamb's remarks — Oct. 24, 2015).

Some context:  Gödel's Platonic realism vs. Hilbert's axiomatics
in remarks by Manuel Alfonseca on June 7, 2018. (See too remarks
in this journal on that date, in posts tagged "Road to Hell.")

Wednesday, November 7, 2018

Defense Against the Dark Arts

Filed under: General — m759 @ 6:19 AM

"What is the Necropastoral?" was published on April 29th, 2014.
Also on April 29th, 2014 —

See as well Christianity and Culture  by T. S. Eliot.

Friday, November 2, 2018

Elevation of the Hostess

Filed under: General — m759 @ 1:27 AM

Or:  Netflix and Chill on All Souls' Day

Midrash from Huffington Post  (click to enlarge) —

Wednesday, July 11, 2018

Plato and Paradigms, Revisited

Filed under: General — m759 @ 12:00 AM

"Plato thought nature but a spume that plays
Upon a ghostly paradigm of things"

— W. B. Yeats, "Among School Children"

Friday, October 13, 2017

Speak, Memra

Filed under: General,Geometry — Tags: — m759 @ 11:59 AM

The above was suggested by a Log24 review of October 13, 2002,
which in turn suggested a Log24 search for Carousel that yielded
(from Bloomsday Lottery) —

See as well Asimov's "prime radiant," and an illustration
of the number 13 as a radiant prime

"The Prime Radiant can be adjusted to your mind,
and all corrections and additions can be made
through mental rapport. There will be nothing to
indicate that the correction or addition is yours.
In all the history of the Plan there has been no
personalization. It is rather a creation of all of us 
together. Do you understand?"  

"Yes, Speaker!"

— Isaac Asimov, 
    Second Foundation , Ch. 8: Seldon's Plan

"Before time began, there was the Cube."
— Optimus Prime

See also Transformers in this journal.

Monday, October 9, 2017

¿Águila o Sol?

Filed under: General — Tags: — m759 @ 9:00 PM

Saturday, September 30, 2017

Paradoxes for Oxymorons

Filed under: General — m759 @ 6:00 PM

"This poem is concerned with language on a very plain level."

— "Paradoxes and Oxymorons" by John Ashbery,
quoted here in the post Take Your Pick of Dec. 16, 2011.

"The problem is a paradox of the veridical  type,
because the correct result (you should switch doors)
is so counterintuitive it can seem absurd,
but is nevertheless demonstrably true."

— Wikipedia on the Monty Hall problem.

Related material —

Saturday, July 29, 2017

MSRI Program

Filed under: General,Geometry — Tags: — m759 @ 8:29 PM

"The field of geometric group theory emerged from Gromov’s insight
that even mathematical objects such as groups, which are defined
completely in algebraic terms, can be profitably viewed as geometric
objects and studied with geometric techniques."

— Mathematical Sciences Research Institute, 2016:

Geometric Group theory at MSRI (pronounced 'Misery')

See also some writings of Gromov from 2015-16:

For a simpler example than those discussed at MSRI
of both algebraic and geometric techniques applied to
the same group, see a post of May 19, 2017,
"From Algebra to Geometry." That post reviews
an earlier illustration —

For greater depth, see "Eightfold Cube" in this journal.

Wednesday, July 12, 2017

Say Cheese

Filed under: General — m759 @ 10:23 PM

For Indiana Langdon and the Harvard Foundation

Friday, July 7, 2017

Psycho History

Filed under: General — m759 @ 6:00 PM

The title was suggested by the term "psychohistory" in
the Foundation  novels of Isaac Asimov. See the previous post.

See also a 2010 New York Times  review of
DeLillo's novel Point Omega . The review is titled,
without any other reference to L'Engle's classic tale
of the same name, "A Wrinkle in Time."

IMAGE- NY Times headline 'A Wrinkle in Time' with 24 Hour Psycho and Point Omega scene

Related material: The Crosswicks Curse.

A Prime Radiant for Krugman

Filed under: General — Tags: — m759 @ 6:00 AM

Paul Krugman:
Asimov's Foundation  novels grounded my economics

In the Foundation  novels of Isaac Asimov

"The Prime Radiant can be adjusted to your mind, and all
corrections and additions can be made through mental rapport.
There will be nothing to indicate that the correction or addition
is yours. In all the history of the Plan there has been no
personalization. It is rather a creation of all of us together.
Do you understand?"  

"Yes, Speaker!"

— Isaac Asimov, Second Foundation , Ch. 8: Seldon's Plan

"Before time began, there was the Cube."

See also Transformers in this journal.

Wednesday, June 7, 2017

Anywhere in Years

Filed under: General — m759 @ 10:27 PM

" In 1965, Mr. Simmons, an incisive, erudite reviewer and essayist,
won a William Faulkner Foundation Award for Powdered Eggs  [ 1964 ],
recognized as a notable first novel. (He wryly called it his
'64th first novel.') The Boston Globe  said it was 'certainly among
the outstanding fictions of the ′60s.' [ Later, in 1971,* ] The novelist
Harry Crews heralded him as 'one of the finest comic voices to appear
anywhere in years.' "

— Sam Roberts in New York Times  obituary this evening

See also Harry Crews in this journal.

Roberts says Simmons also wrote "a savage sendup of  The New York Times
Book Review
, where he had worked as an editor for three decades."

Some not-so-savage related material —

'Watchmen'-like art in the Feb. 21, 2016, NY Times Book Review

* "Anywhere in years" — From http://www.nytimes.com/1971/11/21/archives/
an-oldfashioned-darling-by-charles-simmons- 202-pp-new-york-coward.html

Wednesday, January 18, 2017

An Associative Function …

Filed under: General — m759 @ 2:02 PM

Quoted here on December 16, 2006

'An associative function' in cubist collage and in Joyce's Ulysses, in a paper by Archie K. Loss

See also …

The date  of the "Seconds" review above, 16 Dec. 2006, was 
the reason for the requotation in the first paragraph above.

Sunday, December 4, 2016

London Recessional

Filed under: General — Tags: — m759 @ 12:01 PM

Jack London on Kipling —

Also for "Recessional."

Sunday, November 13, 2016


Filed under: General — m759 @ 11:30 PM

Sewell is supposedly modeled on literary critic R. P. Blackmur.
For a quotation from Blackmur, see a post of June 1, 2006.

Friday, October 7, 2016


Filed under: General — Tags: — m759 @ 11:27 AM

The Paz quote below is from the last chapter
of his book, titled "The Dialectic of Solitude."

The phrase "dialectic of solitude" has been applied also to a 1967
book by the Colombian writer Gabriel García Márquez:

The conclusion of One Hundred Years of Solitude ,
a 1967 novel by Gabriel García Márquez —

"He was so absorbed that he did not feel the second surge of wind either as its cyclonic strength tore the doors and windows off their hinges, pulled off the roof of the east wing, and uprooted the foundations. Only then did he discover that Amaranta Úrsula was not his sister but his aunt, and that Sir Francis Drake had attacked Riohacha only so that they could seek each other through the most intricate labyrinths of blood until they would engender the mythological animal that was to bring the line to an end. Macondo was already a fearful whirlwind of dust and rubble being spun about by the wrath of the biblical hurricane when Aureliano skipped eleven pages so as not to lose time with facts he knew only too well, and he began to decipher the instant that he was living, deciphering it as he lived it, prophesying himself in the act of deciphering the last page of the parchments, as if he were looking into a speaking mirror. Then he skipped again to anticipate the predictions and ascertain the date and circumstances of his death. Before reaching the final line, however, he had already understood that he would never leave that room, for it was foreseen that the city of mirrors (or mirages) would be wiped out by the wind and exiled from the memory of men at the precise moment when Aureliano Babilonia would finish deciphering the parchments, and that everything written on them was unrepeatable since time immemorial and forever more, because races condemned to one hundred years of solitude did not have a second opportunity on earth."

Update of Saturday, October 8:

I do not recommend taking very seriously the work of Latin American leftists
(or American academics) who like to use the word "dialectic."

A related phrase does, however, have a certain mystic or poetic charm,
as pointed out by Wikipedia —

"Unity of opposites is the central category of dialectics,
and it is viewed sometimes as a metaphysical concept,
a philosophical concept or a scientific concept."

See also Bullshit Studies.

Sunday, September 25, 2016

Introduction to Pragmatism

Filed under: General — Tags: — m759 @ 7:29 AM

Stanford Encyclopedia of Philosophy
on the origins of Pragmatism:

"Pragmatism had been born in the discussions at
a ‘metaphysical club’ in Harvard around 1870
(see Menand…*). Peirce and James participated
in these discussions along with some other philosophers
and philosophically inclined lawyers. As we have
already noted, Peirce developed these ideas in his
publications from the 1870s."

From "How to Make Our Ideas Clear,"
by Charles Sanders Peirce in 1878 —

"The very first lesson that we have a right to demand that logic shall teach us is, how to make our ideas clear; and a most important one it is, depreciated only by minds who stand in need of it. To know what we think, to be masters of our own meaning, will make a solid foundation for great and weighty thought. It is most easily learned by those whose ideas are meagre and restricted; and far happier they than such as wallow helplessly in a rich mud of conceptions. A nation, it is true, may, in the course of generations, overcome the disadvantage of an excessive wealth of language and its natural concomitant, a vast, unfathomable deep of ideas. We may see it in history, slowly perfecting its literary forms, sloughing at length its metaphysics, and, by virtue of the untirable patience which is often a compensation, attaining great excellence in every branch of mental acquirement. The page of history is not yet unrolled which is to tell us whether such a people will or will not in the long-run prevail over one whose ideas (like the words of their language) are few, but which possesses a wonderful mastery over those which it has. For an individual, however, there can be no question that a few clear ideas are worth more than many confused ones. A young man would hardly be persuaded to sacrifice the greater part of his thoughts to save the rest; and the muddled head is the least apt to see the necessity of such a sacrifice. Him we can usually only commiserate, as a person with a congenital defect. Time will help him, but intellectual maturity with regard to clearness comes rather late, an unfortunate arrangement of Nature, inasmuch as clearness is of less use to a man settled in life, whose errors have in great measure had their effect, than it would be to one whose path lies before him. It is terrible to see how a single unclear idea, a single formula without meaning, lurking in a young man's head, will sometimes act like an obstruction of inert matter in an artery, hindering the nutrition of the brain, and condemning its victim to pine away in the fullness of his intellectual vigor and in the midst of intellectual plenty. Many a man has cherished for years as his hobby some vague shadow of an idea, too meaningless to be positively false; he has, nevertheless, passionately loved it, has made it his companion by day and by night, and has given to it his strength and his life, leaving all other occupations for its sake, and in short has lived with it and for it, until it has become, as it were, flesh of his flesh and bone of his bone; and then he has waked up some bright morning to find it gone, clean vanished away like the beautiful Melusina of the fable, and the essence of his life gone with it. I have myself known such a man; and who can tell how many histories of circle-squarers, metaphysicians, astrologers, and what not, may not be told in the old German story?"

Peirce himself may or may not have been entirely successful
in making his ideas clear.  See Where Credit Is Due  (Log24, 
June 11, 2016) and the Wikipedia article Categories (Peirce).

* Menand, L., 2001. The Metaphysical Club A Story of
Ideas in America
, New York:  Farrar, Straus and Giroux

Friday, September 16, 2016

A Counting-Pattern

Filed under: General,Geometry — Tags: , — m759 @ 10:48 AM

Wittgenstein, 1939

Dolgachev and Keum, 2002

IMAGE- Dolgachev and Keum, coordinatization of the 4x4 array in 'Birational Automorphisms of Quartic Hessian Surfaces,' AMS Transactions, 2002

For some related material, see posts tagged Priority.

Monday, July 25, 2016

The Retinoid Self

Filed under: General,Geometry — m759 @ 1:10 PM

Trehub - 'Self as the neuronal origin of retinoid space'

"… which grounds the self" . . .

Popular Mechanics  online today

"Verizon exec Marni Walden seemed to
indicate Mayer's future may still be up in the air."

See also 5×5 in this  journal —

IMAGE- Right 3-4-5 triangle with squares on sides and hypotenuse as base

"If you have built castles in the air, 
your work need not be lost;
that is where they should be.
Now put the foundations under them.”

— Henry David Thoreau

Thursday, July 7, 2016

About Nothing

Filed under: General — Tags: — m759 @ 9:16 PM

"For the listener, who listens in the snow, 
And, nothing himself, beholds 
Nothing that is not there and the nothing that is."

Wallace Stevens

See as well A Riddle for Davos.

Friday, July 1, 2016

Dead Poet

Filed under: General — m759 @ 2:56 PM
BBC News today ("4 hours ago" at 2:50 PM ET) —

Poet Sir Geoffrey Hill dies aged 84

British poet Sir Geoffrey Hill has died aged 84,
his wife has confirmed.

Alice Goodman said her husband died "suddenly,
and without pain or dread" on Thursday evening.

Sir Geoffrey was the Professor of Poetry at
Oxford University until last year, and best known for
Mercian Hymns , his 1971 collection of prose poems. . . .

Poet laureate Carol Ann Duffy told The Guardian :
"He was, in poetry, a saint and a warrior
who never gave an inch in his crusade to reach poetic truth."

See also Geoffrey Hill at Poetry Foundation and in this journal.

Thursday, June 23, 2016

Raiders of the Lost Code

Filed under: General — m759 @ 11:00 AM

From a web page

Breaking the Code of the Archetypal Self:
An Introductory Overview of the Research Discoveries
Leading to Neo-Jungian Structural Psychoanalysis

Dr. Moore will introduce his research and discoveries
with regard to the deep structures of the Self.
Tracing the foundations in the tradition of Jung’s
affirmation of the collective unconscious, Moore
will present his “decoding of the Diamond Body,”
a mapping of the deep structures of the Great Code
of the psyche. . . .

From the same web site

Googling "Jung" + "Diamond Body" shows that
Moore's terminology differs from Jung's.
The octahedron that Moore apparently associates
with his "diamond body" was discussed by Jung
in a different context. See selections from Ch. 14
of Jung's Aion
 "The Structure and Dynamics of the Self."

Dr. Moore appears as well in the murder-suicide story 
of last night's 11:18 PM ET post.

For the relevance of Aion  to "deep structures,"
see Jung + Diamond + Structure in this  journal
and, more specifically, "Deep  Structure."

Thursday, May 12, 2016

But Seriously …

Filed under: General — m759 @ 7:31 PM

Google today released on GitHub an English parser,
Parsey McParseface .  From Google Research Blog

"Today, we are excited to share the fruits of our research
with the broader community by releasing SyntaxNet,
an open-source neural network framework implemented in 
TensorFlow that provides a foundation for 
Natural Language Understanding (NLU) systems.
Our release includes all the code needed to train new
SyntaxNet models on your own data, as well as 
Parsey McParseface , an English parser that we have
trained for you and that you can use to analyze English text."

"While the accuracy is not perfect, it’s certainly high enough
to be useful in many applications. The major source of errors
at this point are examples such as the prepositional phrase
attachment ambiguity described above, which require real
world knowledge (e.g. that a street is not likely to be located
in a car) and deep contextual reasoning. Machine learning
(and in particular, neural networks) have made significant
progress in resolving these ambiguities. But our work is still
cut out for us: we would like to develop methods that can
learn world knowledge and enable equal understanding of
natural language across all  languages and contexts."

But seriously

For some historical background, see (for instance) a book by
Ekaterina Ovchinnikova —

Integration of World Knowledge for
Natural Language Understanding
Atlantis Press, Springer, 2012.

A PDF of Chapter 2, "Natural Language Understanding
and World Knowledge," is available for download.

The philosophical background is the distinction between
syntax  and semantics . See (for instance)

Gian-Carlo Rota on Syntax and Semantics

Wednesday, May 11, 2016

By Diction Possessed

Filed under: General — Tags: — m759 @ 12:00 PM

Continued from Saturday, May 7, 2016 .

From an obituary in yesterday evening's online New York Times —

"I was writing plays, one-acters, about musicians
who were speakers of the idiom I loved most:
black American male speech, full of curse words,"
he wrote in an autobiographical essay. . . .

The obituary is for a poet who reportedly died on Saturday, May 7.

This  journal on that day ("By Diction Possessed") recalled the death
(on Valentine's Day 2015) of an English actor who was the voice of
the Ring in two of the "Lord of the Rings" films —

Backstory from Wikipedia — See Black Speech —

"The only example of 'pure' Black Speech is
the inscription upon the One Ring . . .

One Ring to rule them all,
One Ring to find them,
One Ring to bring them all
And in the darkness bind them. 

Thursday, April 14, 2016

Strange Awards

Filed under: General,Geometry — Tags: , — m759 @ 10:38 AM

From a review of a play by the late Anne Meara* —

"Meara, known primarily as an actress/comedian
(half of the team of Stiller & Meara, and mother of
Ben Stiller), is also an accomplished writer for the
stage; her After Play  was much acclaimed….
This new, more ambitious piece starts off with a sly
send-up of awards dinners as the late benefactor of
a wealthy foundation–the comically pixilated scientist
Herschel Strange (Jerry Stiller)–is seen on videotape.
This tape sets a light tone that is hilariously
heightened when John Shea, as Arthur Garden,
accepts the award given in Strange's name." 

Compare and contrast —

A circular I Ching

I of course prefer the Galois I Ching .

* See the May 25, 2015, post The Secret Life of the Public Mind.

Thursday, April 7, 2016

Plan 9 Continues (Very Slowly)

Filed under: General — m759 @ 1:00 PM

From the American Mathematical Society today —

Verena Huber-Dyson (1923-2016) 
Thursday April 7th 2016

Huber-Dyson, who worked in group theory and logic, died March 12 at the age of 92. She was born to Swiss parents in Naples and grew up in Athens but moved with her family to Switzerland in 1940 because of the war. Huber-Dyson received her PhD from the University of Zurich in 1947 and moved to the U.S. in 1948 for a post-doctoral fellowship at the Institute for Advanced Study. She held positions at many universities around the world and taught courses on the foundation of mathematics as a philosophy professor at the University of Calgary from 1973 to 1988. Her son, George Dyson, said that her last words were, "This will all go smoothly. Let's get going." She was an AMS member since 1949. Read more about her life.      

See also Huber-Dyson in this  journal.

Monday, March 28, 2016

Easter, 2016

Filed under: General — m759 @ 1:00 PM

"Minute by minute they live:
The stone's in the midst of all."
— "Easter, 1916"

A Philosophers' Stone
St. Patrick's Day, 2016

Tuesday, February 2, 2016

Depth Haiku

Filed under: General — m759 @ 8:30 PM

From yesterday's 2 PM post

From "Inception" —

Paraphrase of remarks by "Inception" director Christopher Nolan
at Princeton on June 1, 2015 —

"If you have built castles in the air, 
your work need not be lost;
that is where they should be.
Now put the foundations under them.”

— Henry David Thoreau

Tuesday, January 26, 2016

Ground Plan

Filed under: General — m759 @ 2:08 AM

The New York Times , reporting yesterday on the death of
a distinguished expert in the field of artificial intelligence,
said that he "laid the foundation for the field."

Related material:

"A Bad Case of Mixed Metaphors:
Psychiatry, Law, Politics, Society,
and Ezra Pound"

by Arnold M. Ludwig, 
American Journal of Psychotherapy
2000 Winter; 54(1): 116-117

From that paper
" the conceptual foundation for the field
continues to be primitive."

See also, in this  journal, The Source (Oct. 4, 2014).
An image from that post

Ground plan for a game of
Noughts and Crosses

Wednesday, January 20, 2016

Fringe Physics and Beyond

Filed under: General,Geometry — m759 @ 11:00 AM

"One day not long ago Oppenheimer stalked
up and down his office and divulged some
startling new discoveries about the 15 fundamental
particles of which the universe is made….

physicists today are wondering if these particles
are themselves actually the final, stark, immutable
and indivisible foundation stones of the universe
that until now they have been thought to be."

—Lincoln Barnett in LIFE magazine,
    Oct. 10, 1949, page 122

Fringe Physics

" astrophysics limits the number of fundamental particles to 15…."

— Franklin Potter at FQXi.org on Sep. 27, 2009

"I agree there can't be more than 15 fundamental particles."

— Lawrence B. Crowell at FQXi.org on Sep. 29, 2009


There are, at any rate, 15 "final, stark, immutable* and indivisible*
foundation stones" (namely, 15 points ) of the finite projective
space PG(3,2). See Symplectic  in this journal.

For related physics, see posts tagged Dirac and Geometry.

* Update of Jan. 21, 2016 — I was carried away by Barnett's
   powerful rhetoric. These adjectives are wrong.

Monday, November 2, 2015

The Devil’s Offer

Filed under: General,Geometry — Tags: — m759 @ 11:09 AM

This is a sequel to the previous post and to the Oct. 24 post
Two Views of Finite Space.  From the latter —

" 'All you need to do is give me your soul:  
give up geometry and you will have this 
marvellous machine.' (Nowadays you
can think of it as a computer!) "

George Boole in image posted on All Souls' Day 2015

Saturday, October 31, 2015

Weaving World…

Filed under: General,Geometry — m759 @ 6:00 PM


Addendum —

      See also Symplectic Structure 
      and Stevens's Rock.

Wednesday, August 5, 2015

At Zero Dark Thirty…

Filed under: General — Tags: — m759 @ 12:30 AM

"I sing the body electric." (See previous post.)

Wednesday, July 8, 2015

In Memoriam: Stupski

Filed under: General — m759 @ 11:30 AM

From the date of Lawrence Stupski's death —

See as well a search in this journal for "Foundation."

Design Thinking

Filed under: General — m759 @ 11:14 AM

This post was suggested in part by last night's post
of 11:14 PM ET, Southern Charm, and by a post
of 11/14 last year, Another Opening, Another Show.

See also Design Thinking at Wikipedia and the following
two quotations —

Dr. Gerrita Postlewait's contract for Superintendent
of Charleston County Schools was approved and
signed in a meeting with school board members 
Wednesday morning, a school system official says….
From 2006 to 2013, she was the chief K-12 officer for
the Stupski Foundation, a San Francisco-based
education reform nonprofit. [See related page.]

Chris Tebben, executive director of Grantmakers for
Education, says the [Stupski] foundation was among
the first to consider how the problem-solving approach
known as “design thinking” could play a role in improving

Related cinematic remarks:  Robot Overlords (now on-demand).

Friday, June 12, 2015

Dickens for Child Buyers

Filed under: General — m759 @ 1:00 AM

"              … yet the dread
Of dying, and being dead,
Flashes afresh to hold and horrify."

— Philip Larkin, "Aubade"

From tonight's New York Times  obituaries —

See as well "Child Buyer" in this journal.

Tuesday, June 2, 2015

Miniature Lowry

Filed under: General,Geometry — m759 @ 7:59 AM

A review describes the main character in the 1981 novel
The Knife in My Hands —

"His other peculiar characteristic is a fascination, developed during
his university years, with the Cabala and, like a miniature Lowry,*
he spends much of his time wandering drunkenly through its
labyrinthine mysteries."

AA motto:  "Principles before personalities."

* Related material:  Back to the Real.

Wednesday, May 20, 2015


Filed under: General — m759 @ 1:00 AM

Templeton reportedly died on Saturday, May 16, 2015.

Wednesday, May 6, 2015

Narrative Metaphysics

Filed under: General — m759 @ 11:01 PM

(Continued from Dec. 13, 2014.)

David Lavery's enthusiasm today for the Marvel Comics
"Infinity Stones" suggests a review of The Foundation Stone
mentioned in the post Narrative Metaphysics of 12/13/2014.

See as well "Many Dimensions" in this journal.

Saturday, April 25, 2015

Ageometretos Medeis Eisito

Filed under: General — m759 @ 10:30 AM

Wednesday, April 1, 2015


Filed under: General,Geometry — Tags: , , — m759 @ 7:59 PM

Continued from yesterday, the date of death for German
billionaire philanthropist Klaus Tschira —

For Tschira in this journal, see Stiftung .

For some Würfel  illustrations, see this morning's post
Manifest O.  A related webpage —

Thursday, March 19, 2015

Midnight in the Garden continues…

Filed under: General — m759 @ 12:00 AM

See also Icon and a fresh New York Times  obituary.

Happy birthday to Jill Abramson.

Friday, March 13, 2015

Mathieu Moonshine

Filed under: General — m759 @ 1:26 PM

(Continued from yesterday's "earlier references" link.)

Yesterday at the Simons Foundation's Quanta Magazine :

See also earlier Log24 references to Mathieu moonshine .
I do not know the origin of this succinct phrase, taken from
an undated web page of Anne Taormina.

Friday, January 30, 2015

The Relaxed Field*

Filed under: General — m759 @ 7:00 PM

In memory of a dead poet —

"Relax," said the night man.
"We are programmed to receive."

* A phrase from a new book by mathematician
  Michael Harris, Mathematics without Apologies .

Monday, December 15, 2014

May We?

Filed under: General — m759 @ 11:29 AM

John Burt Foster Jr. in Nabokov's Art of Memory and
European Modernism
  (Princeton U. Press, 1993, p. 224) —

At the time of The Waste Land , in a comment on
Joyce's Ulysses  that influenced many later definitions
of modernism in the English-speaking world, Eliot
announced, "instead of narrative method, we may
now use the mythical method."13

May we? … Further details —


T. S. Eliot, "'Ulysses,' Order and Myth,"
in The Dial , LXXV, No. 5 (Nov. 1923),
pp. 480-83,

the last two paragraphs:

It is here that Mr Joyce’s parallel use of the Odyssey  has a great importance. It has the importance of a scientific discovery. No one else has built a novel upon such a foundation before: it has never before been necessary. I am not begging the question in calling Ulysses  a novel; and if you call it an epic it will not matter. If it is not a novel, that is simply because the novel is a form which will no longer serve; it is because the novel, instead of being a form, was simply the expression of an age which had not sufficiently lost all form to feel the need of something stricter. Mr Joyce has written one novel – the Portrait ; Mr Wyndham Lewis has written one novel – Tarr . I do not suppose that either of them will ever write another “novel.” The novel ended with Flaubert and with James. It is, I think, because Mr Joyce and Mr Lewis, being “in advance” of their time, felt a conscious or probably unconscious dissatisfaction with the form, that their novels are more formless than those of a dozen clever writers who are unaware of its obsolescence.

In using the myth, in manipulating a continuous parallel between contemporaneity and antiquity, Mr Joyce is pursuing a method which others must pursue after him. They will not be imitators, any more than the scientist who uses the discoveries of an Einstein in pursuing his own, independent, further investigations. It is simply a way of controlling, of ordering, of giving a shape and a significance to the immense panorama of futility and anarchy which is contemporary history. It is a method already adumbrated by Mr Yeats, and of the need for which I believe Mr Yeats to have been the first contemporary to be conscious. It is a method for which the horoscope is auspicious. Psychology (such as it is, and whether our reaction to it be comic or serious), ethnology, and The Golden Bough have concurred to make possible what was impossible even a few years ago. Instead of narrative method, we may now use the mythical method. It is, I seriously believe, a step towards making the modern world possible for art, toward that order and form which Mr Aldington so earnestly desires. And only those who have won their own discipline in secret and without aid, in a world which offers very little assistance to that end, can be of any use in furthering this advance.

Monday, December 1, 2014

Change Arises

Filed under: General,Geometry — m759 @ 11:00 AM

Flashback to St. Andrew's Day, 2013 —

Saturday, November 30, 2013

Waiting for Ogdoad

Filed under: Uncategorized — Tags:  — m759 @ 10:30 AM 

Continued from October 30 (Devil's Night), 2013.

“In a sense, we would see that change
arises from the structure of the object.”

— Theoretical physicist quoted in a
Simons Foundation article of Sept. 17, 2013

This suggests a review of mathematics and the
"Classic of Change ," the I Ching .

If the object is a cube, change arises from the fact
that the object has six  faces…

and is the unit cell for the six -dimensional
hyperspace H over the two-element field —

Spaces as Hypercubes

A different representation of the unit cell of
the hyperspace H (and of the I Ching ) —

Wednesday, November 5, 2014

Dark Fields…

Filed under: General — m759 @ 1:06 AM


From the first of previous Log24 posts tagged “Dark Fields”—

“A link in memory of Donald G. Higman,
dead on Feb. 13, 2006,
the day after Lincoln’s birthday:

On the Graphs of Hoffman-Singleton and Higman-Sims.

His truth is marching on.”

See also Foundation Square (October 25, 2014).

Sunday, August 10, 2014


Filed under: General — m759 @ 11:00 AM

From The Mathematics of Language:
10th and 11th  Biennial Conference….
Berlin,  Springer, 2010 —

Creation Myths of Generative Grammar
and the Mathematics of Syntactic Structures
by Geoffrey K. Pullum, University of Edinburgh


Syntactic Structures  (Chomsky [6])  is widely believed to have laid
the foundations of a cognitive revolution in linguistic science, and
to have presented (i) the first use in linguistics of powerful new ideas
regarding grammars as generative systems, (ii) a proof that English
was not a regular language, (iii) decisive syntactic arguments against
context-free phrase structure grammar description, and (iv) a
demonstration of how transformational rules could provide a formal
solution to those problems. None of these things are true. This paper
offers a retrospective analysis and evaluation.”

Sunday, July 20, 2014


Filed under: General — m759 @ 11:00 AM

“Tell it slant.” — Emily Dickinson

IMAGE- James Garner, NY Times obits

Saturday, July 19, 2014

Zero Art

Filed under: General — m759 @ 12:00 PM

“Zero is a group of artists founded by Heinz Mack and
Otto Piene….” — Wikipedia

“The title ZERO was the result of months of search
and was finally found more or less by chance.
From the beginning we looked upon the term
not as an expression of nihilism – or a dada-like gag,
but as a word indicating a zone of silence and of
pure possibilities for a new beginning as at the
count-down when rockets take off-  zero is the
incommensurable zone in which the old state turns
into the new.”

Otto Piene, 1964

Alternate Reality

Filed under: General — m759 @ 11:30 AM

(Where Entertainment Is God , continued)

In memory of artist Otto Piene — a news item from last May
at the ZERO Foundation website on an exhibition that closes tomorrow —

Today is the opening of the exhibition ZERO — Zwischen Himmel und Erde
in Friedrichshafen. The Zeppelin Museum is showing wonderful artworks
all related to heaven and earth by various ZERO artists
such as Piene, Mack, Uecker, Klein, Luther, and Manzoni.
ZERO – Zwischen Himmel und Erde
Zeppelin Museum Friedrichshafen
15.05. – 20.07.2014


“Oh, show me the way to the next whiskey bar”
— Song lyric from previous post

“In a technologically advanced 1939, the zeppelin Hindenburg III
arrives in New York City, mooring atop the Empire State Building.”

— Wikipedia on the first scene of the 2004 film
Sky Captain and the World of Tomorrow

Friday, July 4, 2014


Filed under: General,Geometry — m759 @ 9:29 AM

(Continued from yesterday’s noon post, from  “Block That Metaphor,”
and from “Mystery Box III: Inside, Outside“)

“In one corner are the advocates of the Common Core,
led by the Bill and Melinda Gates Foundation, which
helped develop the standards and has defended them
against efforts by some states to roll them back. In the
challengers’ corner, a lineup of foundations and
philanthropists…. Other funders in the opponents’ corner
read like a ‘who’s who’ of well-heeled conservative
philanthropists, including Pittsburgh media magnate
Richard Mellon Scaife….”

— “Meet the Funders Fighting the Common Core,”
from Inside Philanthropy , Feb. 10, 2014

Scaife reportedly died this morning.

Thursday, July 3, 2014

Gates and Windows:

Filed under: General — m759 @ 12:00 PM

The Los Alamos Vision

“Gates said his foundation is an advocate for the Common Core State Standards
that are part of the national curriculum and focus on mathematics and language
arts. He said learning ‘needs to be on the edge’ where it is challenging but not
too challenging, and that students receive the basics through Common Core.

‘It’s great to teach other things, but you need that foundation,’ he said.”

— T. S. Last in the Albuquerque Journal , 12:05 AM Tuesday, July 1, 2014

See also the previous post (Core Mathematics: Arrays) and, elsewhere
in this journal,

“Eight is a Gate.” — Mnemonic rhyme:

Thursday, May 1, 2014


Filed under: General — m759 @ 1:44 AM

(Continued from Tuesday morning)

IMAGE- Tom Cruise as Jack Reacher

… Or what’s a heaven for?

Thursday, April 10, 2014

Change Arises (continued)

Filed under: General — m759 @ 11:00 PM

In memory of Lucia Eames, who reportedly died
on April 1, 2014:

“… Walter Gropius was her professor ….”

See also in this journal Gropius and the April 1 posts.

Related material: “As a little child.”

Wednesday, April 2, 2014

Change Arises

Filed under: General — Tags: — m759 @ 9:00 AM

IMAGE- Search for the source of the quotation 'Change arises from the structure of the object'

For a different view of change arising, click on the tag above.

Wednesday, March 26, 2014

Sinai Prize

Filed under: General — m759 @ 1:00 PM

Yakov G. Sinai today won the 2014 Abel Prize.
Earlier, he won the Wolf Prize.

Wolf Foundation press release quoted in the March 1997
Notices of the American Mathematical Society —

On Sinai —

“He is generally recognized as the world leader
in the mathematics of statistical physics.”

This afternoon’s New York Lottery:  813 and 1857.

Unrelated remarks:  813 and  1857.

Friday, February 14, 2014


Filed under: General,Geometry — m759 @ 12:00 PM

“… the object sets up a kind of 
 frame or space or field 
 within which there can be epiphany.”

Charles Taylor

A frame or space or field —

IMAGE- The ninefold square

Related material —

Star Wars (January 11, 2014),

The Lyche Gate Asterisk , from 10:31 AM ET on May 22, 2010,
the date of Martin Gardner's death —

Image-- The Case of the Lyche Gate Asterisk

— and the March 2014 issue of the
Notices of the American Mathematical Society  —

See as well Epiphany 2014 (Jan. 6) in this journal and the
March Notices  on the Shaw prize —

"Established under the auspices of Run Run Shaw
in November 2002, the prize is managed and
administered by the Shaw Prize Foundation
based in Hong Kong." 

Wednesday, January 15, 2014

Entertainment Theory

Filed under: General,Geometry — Tags: — m759 @ 1:00 PM

From "Entertainment," a 1981 story by M. A. Foster—

"For some time, Cormen had enjoyed a peculiar suspicion, which he had learned from his wanderings around the city, and cultivated with a little notebook, in which he had made a detailed series of notes and jottings, as well as crude, but effective, charts and maps of certain districts. 'Cormen's Problem,' as it was known, was familiar to the members of the circle in which he moved; in fact, if he had not been so effective with his productions and so engaging in his personality, they might have considered him a bore.

It seemed, so the suspicion went, that the city was slowly shrinking, as evidenced by abandoned districts along the city edges. Beyond the empty houses were ruins, and beyond that, traces of foundations and street lines. Moreover, it had recently dawned on him that there were no roads out of the city, although there were no restraints. One hardly noticed this—it was the norm. But like many an easy assumption, once broken it became increasingly obvious.

Cormen's acquaintances were tolerant of his aberration, but generally unsympathetic. What he needed was proof, something he could demonstrate in black and white—and color if required. But the city was reluctant, so it appeared, to give up its realities so easily. The Master Entertainment Center, MEC, would not answer direct queries about this, even though it would obediently show him presentations, pictorial or symbolic as he required, of the areas in question. But it was tiresome detail work, in which he had to proceed completely on his own."

Lily Collins in City of Bones  (2013)—

American Folk Art (see August 23, 2011) —

IMAGE- Four Winds quilt block

Art Theory —

IMAGE- The eight Galois quaternions

See as well Ballet Blanc .

Thursday, December 12, 2013

Communion of Saints

Filed under: General — Tags: — m759 @ 11:59 PM

For Hugh Jackman,  St. Gall,  and Delmore Schwartz

IMAGE- Logan and bear in 'The Wolverine' (2013)

"All the saints have powers."

Saturday, December 7, 2013

Shadow of a Lie

Filed under: General — m759 @ 1:20 AM

A recent addition to Barry Mazur's home page

"December 1, 2013: Here are rough notes for
a short talk entitled The Faces of Evidence
(in Mathematics)
([PDF]) to be given at the
Cambridge Scientific Club, Dec. 5 2013."

The PDF link does not work, but some earlier remarks by
Mazur on this topic have been published elsewhere:

IMAGE- 'Shadows of Evidence,' by Barry Mazur

Related material:

The Proof and the Lie (St. Andrew's Day, 2003), and
a recent repetition of the lie in Wikipedia:

"Around 1955, Japanese mathematicians Goro Shimura 
Yutaka Taniyama observed a possible link between
two apparently completely distinct, branches of mathematics, 

elliptic curves and modular forms."

This statement, from the article on Algebraic number theory,
was added on Oct. 22, 2013 by one "Brirush," apparently a
Temple University postdoctoral researcher, in what he rightly
called a "terrible history summary."

Saturday, November 30, 2013

Waiting for Ogdoad

Filed under: General,Geometry — Tags: — m759 @ 10:30 AM

Continued from October 30 (Devil's Night), 2013.

“In a sense, we would see that change
arises from the structure of the object.”

— Theoretical physicist quoted in a
Simons Foundation article of Sept. 17, 2013

This suggests a review of mathematics and the
"Classic of Change ," the I Ching .

The physicist quoted above was discussing a rather
complicated object. His words apply to a much simpler
object, an embodiment of the eight trigrams underlying
the I Ching  as the corners of a cube.

The Eightfold Cube and its Inner Structure

See also

(Click for clearer image.)

The Cullinane image above illustrates the seven points of
the Fano plane as seven of the eight I Ching  trigrams and as
seven natural ways of slicing the cube.

For a different approach to the mathematics of cube slices,
related to Gauss's composition law for binary quadratic forms,
see the Bhargava cube  in a post of April 9, 2012.

Friday, November 29, 2013

Odd Facts

Filed under: General — m759 @ 7:01 PM

"These are odd facts…." — G. H. Hardy,
quoted in the previous post, "Centered"

Other odd facts:

If is odd, then the object at the center  
of the n×n  square is a square.
Similarly for the n×n×n  cube.

Related meditation:

“In a sense, we would see that change
arises from the structure of the object,” he said.
“But it’s not from the object changing.
The object is basically timeless.”

— Theoretical physicist quoted in a
Simons Foundation article of Sept. 17, 2013,
"A Jewel at the Heart of Quantum Physics"

See also "My God, it's full of… everything."

Monday, November 25, 2013


Filed under: General — m759 @ 11:22 AM

Ben Brantley reviewing a show by the X-Men patriarchs
that opened on Sunday:

"This isn’t just a matter of theatergoers chuckling
to show that they’re smart and cultured and had
damn well better be having a good time after
forking out all that money…."

I prefer reality (which includes the life of Fred Kavli:

See also Saturday's posts Chess and Frame Tale.

Whether the patriarch Kavli, pictured above, is now having
a good time, I do not know. I hope so.

Saturday, November 23, 2013


Filed under: General — m759 @ 9:29 AM

Norwegian, 22, Takes World Chess Title

Quoted here on Thursday, the date of Kavli's death:

Herbert Mitgang's New York Times  
obituary of Cleanth Brooks

"The New Critics advocated close reading of literary texts
and detailed analysis, concentrating on semantics, meter,
imagery, metaphor and symbol as well as references to
history, biography and cultural background."

See also Steiner, Chess, and Death.

Monday, October 7, 2013

The Book of Abraham

Filed under: General — Tags: — m759 @ 12:00 AM

On Abraham Nemeth, the developer of Braille
for blind students of mathematics —

"… he began tinkering with the six-dot cell
that is the foundation of Braille."

For a different six-dot cell, see Nocciolo .

"Throughout his life, he dedicated much of his
spare time to creating Braille versions of Jewish
texts, including helping to proofread a Braille
Hebrew Bible in the 1950s." — Nemeth's obituary

Those who prefer entertainment may consult  The Book of Eli.

Friday, September 6, 2013


Filed under: General,Geometry — Tags: — m759 @ 9:15 PM

"A vast space that travels down to the bedrock
upon which the towers were built, the museum
winds its way deeper and deeper underground,
taking visitors on a journey to the very bottom."

— The Associated Press in
this evening's Washington Post

This suggests a review of a different sort of

IMAGE- Right 3-4-5 triangle with squares on sides and hypotenuse as base

"If you have built castles in the air, 
your work need not be lost;
that is where they should be.
Now put the foundations under them.”

— Henry David Thoreau

Sunday, July 21, 2013


Filed under: General,Geometry — m759 @ 5:18 AM

This is the weekend for Comic-Con International in San Diego.

The convention includes an art show. (Click above image to enlarge.)

Related material from Norway

IMAGE- The Kavli Prize logo, a Metatron cube

Suggested nominations for a Kavli Prize:

1.  Josefine Lyche's highly imaginative catalog page for
the current Norwegian art exhibition I de lange nætter,
​which mentions her interest in sacred geometry

2.  Sacred Geometry:  Drawing a Metatron Cube

and from San Diego

The Kavli Institutes logo:

IMAGE- Logo of the Kavli institutes

Saturday, July 20, 2013

Vril Chick

Filed under: General — m759 @ 3:15 PM


A Brainstorm  for Jo Lyxe :

Swastika logo of the Kavli Institute for Brain and Mind 
at the University of California, San Diego.

Soundtrack:  Under the Iron Sky .

Thursday, June 20, 2013

ART WARS: Chesterton Thursday

Filed under: General,Geometry — Tags: — m759 @ 8:00 PM

The New York Times  philosophy column "The Stone"
last evening had an essay on art by a sarcastic anarchist,
one Crispin Sartwell

"… whole generations of art lovers have been
trained in modernist dogma, and arts institutions’
access to various forms of state or foundation
support depend on it completely. One goes to
the museum to gasp at stunning works of
incomparable, super-human genius by beings
who are infinitely more exalted and important
than the mere humans staring at their paintings.
That’s why ordinary people staring at a Picasso
(allegedly) experience a kind of transcendence
or re-articulation of their lives and world."

 Cubism Re-Articulated:

  Click image for some backstory.

(IMAGE: Walter Gropius and Froebel's Third Gift,
from a Google image search today)

Background: Cubism in this journal and
Pilate Goes to Kindergarten.

Related material: Chesterton + Thursday in this journal.

Saturday, June 15, 2013

At the Still Point

Filed under: General — m759 @ 1:05 PM

(Continued from yesterday's posts, "Object of Beauty"
and "Amy's Shadow")

A winner of a Nobel Prize for X-ray crystallography stands
at the head of the New York Times  obituary list today.

In memoriam —

X-Ray Vision

"Crystal Engineering in Kindergarten," by Bart Kahr:

"If the reader is beginning to suspect that Froebel’s
philosophy of spiritual crystallography is sometimes
incoherent I can confirm that this is so…."

Click images for some backstories.

Some further background:

The Times  follows yesterday's egregious religious error
with an egregious scientific error:

"The technique developed by Dr. Karle and Herbert A. Hauptman,
called X-ray crystallography, is now routinely used by scientists…."

Karle was reportedly born in 1918, Hauptman in 1917.

Wikipedia on the history of X-ray crystallography:

"The idea that crystals could be used as a 
diffraction grating for X-rays arose in 1912…."

The Nobel Foundation:

"The Nobel Prize in Physics 1914 was awarded to
Max von Laue 'for his discovery of the diffraction of
X-rays by crystals.'"

"The Nobel Prize in Physics 1915 was awarded jointly to
Sir William Henry Bragg and William Lawrence Bragg 
'for their services in the analysis of crystal structure
by means of X-rays.'"

Friday, April 19, 2013

The Large Desargues Configuration

Filed under: General,Geometry — Tags: — m759 @ 9:25 AM

Desargues' theorem according to a standard textbook:

"If two triangles are perspective from a point
they are perspective from a line."

The converse, from the same book:

"If two triangles are perspective from a line
they are perspective from a point."

Desargues' theorem according to Wikipedia
combines the above statements:

"Two triangles are in perspective axially  [i.e., from a line]
if and only if they are in perspective centrally  [i.e., from a point]."

A figure often used to illustrate the theorem,
the Desargues configuration , has 10 points and 10 lines,
with 3 points on each line and 3 lines on each point.

A discussion of the "if and only if" version of the theorem
in light of Galois geometry requires a larger configuration—
15 points and 20 lines, with 3 points on each line
and 4 lines on each point.

This large  Desargues configuration involves a third triangle,
needed for the proof   (though not the statement ) of the
"if and only if" version of the theorem. Labeled simply
"Desargues' Theorem," the large  configuration is the
frontispiece to Volume I (Foundations of Baker's 6-volume
Principles of Geometry .

Point-line incidence in this larger configuration is,
as noted in a post of April 1, 2013, described concisely
by 20 Rosenhain tetrads  (defined in 1905 by
R. W. H. T. Hudson in Kummer's Quartic Surface ).

The third triangle, within the larger configuration,
is pictured below.

IMAGE- The proof of the converse of Desargues' theorem involves a third triangle.

Monday, April 1, 2013

Desargues via Rosenhain

Filed under: General,Geometry — Tags: , — m759 @ 6:00 PM

Background: Rosenhain and Göpel Tetrads in PG(3,2)


The Large Desargues Configuration

Added by Steven H. Cullinane on Friday, April 19, 2013

Desargues' theorem according to a standard textbook:

"If two triangles are perspective from a point
they are perspective from a line."

The converse, from the same book:

"If two triangles are perspective from a line
they are perspective from a point."

Desargues' theorem according to Wikipedia 
combines the above statements:

"Two triangles are in perspective axially  [i.e., from a line]
if and only if they are in perspective centrally  [i.e., from a point]."

A figure often used to illustrate the theorem, 
the Desargues configuration , has 10 points and 10 lines,
with 3 points on each line and 3 lines on each point.

A discussion of the "if and only if" version of the theorem
in light of Galois geometry requires a larger configuration—
15 points and 20 lines, with 3 points on each line 
and 4 lines on each point.

This large  Desargues configuration involves a third triangle,
needed for the proof   (though not the statement ) of the 
"if and only if" version of the theorem. Labeled simply
"Desargues' Theorem," the large  configuration is the
frontispiece to Volume I (Foundations)  of Baker's 6-volume
Principles of Geometry .

Point-line incidence in this larger configuration is,
as noted in the post of April 1 that follows
this introduction, described concisely 
by 20 Rosenhain tetrads  (defined in 1905 by
R. W. H. T. Hudson in Kummer's Quartic Surface ).

The third triangle, within the larger configuration,
is pictured below.

IMAGE- The proof of the converse of Desargues' theorem involves a third triangle.



A connection discovered today (April 1, 2013)—

(Click to enlarge the image below.)

Update of April 18, 2013

Note that  Baker's Desargues-theorem figure has three triangles,
ABC, A'B'C', A"B"C", instead of the two triangles that occur in
the statement of the theorem. The third triangle appears in the
course of proving, not just stating, the theorem (or, more precisely,
its converse). See, for instance, a note on a standard textbook for 
further details.

(End of April 18, 2013 update.)

Update of April 14, 2013

See Baker's Proof (Edited for the Web) for a detailed explanation 
of the above picture of Baker's Desargues-theorem frontispiece.

(End of April 14, 2013 update.)

Update of April 12, 2013

A different figure, from a site at National Tsing Hua University,
shows the three triangles of Baker's figure more clearly:

IMAGE- Desargues' theorem with three triangles, and Galois-geometry version

(End of update of April 12, 2013)

Update of April 13, 2013

Another in a series of figures illustrating
Desargues's theorem in light of Galois geometry:
IMAGE- Veblen and Young 1910 Desargues illustration, with 2013 Galois-geometry version

See also the original Veblen-Young figure in context.

(End of update of April 13, 2013)

Rota's remarks, while perhaps not completely accurate, provide some context
for the above Desargues-Rosenhain connection.  For some other context,
see the interplay in this journal between classical and finite geometry, i.e.
between Euclid and Galois.

For the recent  context of the above finite-geometry version of Baker's Vol. I
frontispiece, see Sunday evening's finite-geometry version of Baker's Vol. IV
frontispiece, featuring the Göpel, rather than the Rosenhain, tetrads.

For a 1986 illustration of Göpel and Rosenhain tetrads (though not under
those names), see Picturing the Smallest Projective 3-Space.

In summary… the following classical-geometry figures
are closely related to the Galois geometry PG(3,2):

Volume I of Baker's Principles  
has a cover closely related to 
the Rosenhain tetrads in PG(3,2)
Volume IV of Baker's Principles 
has a cover closely related to
the Göpel tetrads in PG(3,2) 
(click to enlarge)




Higher Geometry
(click to enlarge)





Wednesday, March 27, 2013

The Barnes School

Filed under: General — m759 @ 9:29 PM

Randy Kennedy in tomorrow's print edition
of The New York Times

Art collector Albert C. Barnes "viewed his foundation
less as a museum than as a school."

Roberta Smith in the New York Times 
print edition of May 18, 2012, on
art arrangements by Albert C. Barnes—

"Barnes’s arrangements are as eye-opening,
intoxicating and, at times, maddening as ever, maybe more so.
They mix major and minor in relentlessly symmetrical patchworks
that argue at once for the idea of artistic genius and the
pervasiveness of talent. Nearly every room is an exhibition
unto itself— a kind of art wunderkammer, or cabinet of curiosities…."

This journal at noon on the same day, May 18, 2012

Balakrishnan's Banners

IMAGE- Eight-limbed star as a problem in combinatorics

See also Brightness at Noon from March 25.

Monday, March 4, 2013

Occupy Galois Space

Filed under: General,Geometry — Tags: — m759 @ 3:00 AM

Continued from February 27, the day Joseph Frank died

"Throughout the 1940s, he published essays
and criticism in literary journals, and one,
'Spatial Form in Modern Literature'—
a discussion of experimental treatments
of space and time by Eliot, Joyce, Proust,
Pound and others— published in
The Sewanee Review  in 1945, propelled him
to prominence as a theoretician."

— Bruce Weber in this morning's print copy
of The New York Times  (p. A15, NY edition)

That essay is reprinted in a 1991 collection
of Frank's work from Rutgers University Press:

See also Galois Space and Occupy Space in this journal.

Frank was best known as a biographer of Dostoevsky.
A very loosely related reference… in a recent Log24 post,
Freeman Dyson's praise of a book on the history of
mathematics and religion in Russia:

"The intellectual drama will attract readers
who are interested in mystical religion
and the foundations of mathematics.
The personal drama will attract readers
who are interested in a human tragedy
with characters who met their fates with
exceptional courage."

Frank is survived by, among others, his wife, a mathematician.

Friday, March 1, 2013

Spinning in Infinity

Filed under: General — Tags: — m759 @ 11:09 PM

(Continued from Jan. 13 and Feb. 19.)

The founder of Graylock Press
died at 96 in Bethesda on Feb. 19:

For some background on the original Bethesda,
see a webpage on Angels in America.

For some background on noted Graylock authors, 
see Pavel Alexandrov.

For deeper background, see a book praised by Freeman Dyson:

"The intellectual drama will attract readers
who are interested in mystical religion
and the foundations of mathematics.
The personal drama will attract readers
who are interested in a human tragedy
with characters who met their fates with
exceptional courage."

Friday, February 8, 2013


Filed under: General,Geometry — Tags: — m759 @ 3:00 PM

A review of the life of physicist Arthur Wightman,
who died at 90 on January 13th, 2013. yields 
the following.

Wightman at Wikipedia:
"His graduate students include
 Arthur Jaffe,  Jerrold Marsden, and Alan Sokal."

"I think of Arthur as the spiritual leader
of mathematical physics and his death
really marks the end of an era."

— Arthur Jaffe in News at Princeton , Jan. 30

Marsden at Wikipedia
"He [Marsden] has laid much of the foundation for
symplectic topology." (Link redirects to symplectic geometry.)

A Wikipedia reference in the symplectic geometry article leads to

Symplectic Geometry Lies at the Very
Foundations of Physics and Mathematics

Mark J. Gotay
Department of Mathematics
University of Hawai‘i

James A. Isenberg
Institute of Theoretical Science and Department of Mathematics
University of Oregon

February 18, 1992


We would like to thank Jerry Marsden and Alan Weinstein
for their comments on previous drafts.

Published in: Gazette des Mathématiciens  54, 59-79 (1992).


"Physics is geometry .  This dictum is one of the guiding
principles of modern physics. It largely originated with
Albert Einstein…."

A different account of the dictum:

The strange term Geometrodynamics 
is apparently due to Wheeler.

Physics may or may not be geometry, but
geometry is definitely not physics.

For some pure geometry that has no apparent 
connection to physics, see this journal
on the date of Wightman's death.

Thursday, October 25, 2012

Bunker Bingo*

Filed under: General — Tags: — m759 @ 11:09 AM

Mathematics and Narrative continues…

IMAGE- In 'Downfall,' Hitler says that 'Steiner's assault will bring it under control.'

Steiner's version of "classical functional analysis"—

"Mein Führer Steiner"

IMAGE- Classical functional analysis according to Moscow State University

* See the story by Kilgore Trout. See also On Linguistic Creation,
   The Matrix of Abraham, and The Thoreau Foundation.

Tuesday, October 23, 2012

Labyrinth 23

Filed under: General — Tags: , , — m759 @ 7:00 PM

The title refers to a search (see below)
suggested by three things—

  1. David Foster Wallace biographer D. T. Max
    "There's a note in one of my files where he says something like,
    'Infinite Jest  was just a means to Mary Karr's end, as it were.' "

  2. "There is a body  on  the cross in my church ." —Mary Karr

  3. A body.

The search Labyrinth 23.

(Within the search results, note particularly the post "The Infinity Point.")

Sunday, October 14, 2012


Filed under: General,Geometry — m759 @ 9:00 PM

"Mathematics is not the rigid and petrifying schema, as the layman so much likes to view it; with it, we rather stand precisely at the point of intersection of restraint and freedom that makes up the essence of man itself."

— A translated remark by Hermann Weyl, p. 136, "The Current Epistemogical Situation in Mathematics" in Paolo Mancosu (ed.) From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s , Oxford University Press, 1998, pp. 123-142, as cited by David Corfield

Corfield once wrote that he would like to know the original German of Weyl's remark. Here it is:

"Die Mathematik ist nicht das starre und Erstarrung bringende Schema, als das der Laie sie so gerne ansieht; sondern wir stehen mit ihr genau in jenem Schnittpunkt von Gebundenheit und Freiheit, welcher das Wesen des Menschen selbst ist."

— Hermann Weyl, page 533 of "Die heutige Erkenntnislage in der Mathematik" (Symposion  1, 1-32, 1925), reprinted in Gesammelte Abhandlungen, Band II  (Springer, 1968), pages 511-542

For some context, see a post of January 23, 2006.

Monday, September 17, 2012

The Count

Filed under: General — Tags: , — m759 @ 11:01 PM

… I saw a shadow
sliding around the ropes
to get at me. The referee
moved it back, and then
went over and picked up the count.
"One!" The fog was clearing.

I rose to a knee,
and at "nine" to my feet.

— Louis Simpson, "The Appointment"

Simpson reportedly died on Holy Cross Day.

That day in this journal—

IMAGE- Log24 posts 'Please Mister Please' and 'Plan 9'

Friday, June 29, 2012

The Uploading (continued)*

Filed under: General — m759 @ 9:29 PM

It Must Be Abstract
It Must Change
It Must Give Pleasure

Parts of a poem by Wallace Stevens

“At that instant he saw, in one blaze of light, an image of unutterable conviction, the reason why the artist works and lives and has his being–the reward he seeks–the only reward he really cares about, without which there is nothing. It is to snare the spirits of mankind in nets of magic, to make his life prevail through his creation, to wreak the vision of his life, the rude and painful substance of his own experience, into the congruence of blazing and enchanted images that are themselves the core of life, the essential pattern whence all other things proceed, the kernel of eternity.”

– Thomas Wolfe, Of Time and the River

      Of Time and the River and the Frogs —

Video uploaded on Jan. 26, 2008, of talk, 'The Lively Kernel,' on object-oriented software

* This post's title refers to the above uploading date—  Jan. 26, 2008.

Friday, February 17, 2012

Pregeometry and Finite Geometry

Filed under: General,Geometry — Tags: — m759 @ 7:35 PM

Today's previous post, on the Feb. 2012 Scientific American
article "Is Space Digital?", suggested a review of a notion
that the theoretical physicist John Archibald Wheeler called
pregeometry .

From a paper on that topic—

"… the idea that geometry should constitute
'the magic building material of the universe'
had to collapse on behalf of what Wheeler
has called pregeometry  (see Misner et al. 1973,
pp. 1203-1212; Wheeler 1980), a somewhat
indefinite term which expresses “a combination
of hope and need, of philosophy and physics
and mathematics and logic” (Misner et al. 1973,
p. 1203)."

— Jacques Demaret, Michael Heller, and
Dominique Lambert, "Local and Global Properties
of the World," preprint of paper published in
Foundations of Science  2 (1): 137-176

Misner, C. W., Thorne, K. S. and Wheeler, J. A.
1973, Gravitation , W.H. Freeman and Company:
San Francisco.

Wheeler, J.A. 1980, "Pregeometry: Motivations
and Prospects," in: Quantum Theory and Gravitation ,
ed. A.R. Marlow, Academic Press: New York, pp. 1-11.

Some related material from pure mathematics—


Click image for further details.

Sunday, January 29, 2012


Filed under: General,Geometry — m759 @ 6:00 PM

Weblog posts of two prominent mathematicians today discussed
what appears to be a revolution inspired by the business practices
of some commercial publishers of mathematics.

See Gowers and Cameron.

My own concern is more with the so-called "Non-Euclidean Revolution"
described by Richard Trudeau in a book of that title (Birkhäuser, 1987).

A 1976 document relevant to the concerns in the Trudeau book—

Though not as well known as another document discussing
"self-evident" truths, Cameron's remarks are also of some
philosophical interest.

They apply to finite  geometry, a topic unknown to Euclid,
but nevertheless of considerable significance for the foundations  
of mathematics.

"The hand of the creative artist, laid upon the major premise,
 rocks the foundations of the world." — Dorothy Sayers

Thursday, January 19, 2012


Filed under: General — m759 @ 8:08 PM

Those impressed by George Steiner's remark on Hegel in the previous post may consult…

(Click to enlarge.)

(The Christian Examiner.  Volume LXXX. New Series, Volume I.  January, March, May, 1866.
New York: James Miller, Publisher, 522, Broadway.  Boston: Walker, Fuller, & Co.

By C. C. Everett, pp. 196-207.

A review of…

The Secret of Hegel, being the Hegelian System in Origin, Principle, Form, and Matter.
By James Hutchinson Sterling. In two volumes.
London: Longman, Green, Longman, Roberts, & Green. 1865. 8vo, 2 vols.)

On Hegel, from the review—

"He starts not from the beginning, but from the heart, of the world.
There never was a time when this pure Being— which, in its
undivided absoluteness, is indistinguishable from nothing;
as pure, unbroken light is indistinguishable from darkness—

was by itself alone; but this absolute Being is yet the foundation
and the groundwork of whatever is."

For more on Hegel's logic, see Marxists.org.

See also Steiner on chess and Lenin in The New Yorker
(September 7, 1968, page 133).

Friday, January 13, 2012


Filed under: General — Tags: — m759 @ 8:28 AM

Click logos for related persons.

IMAGE- Logo of St. Peter's College, Oxford

IMAGE- Logo of St. John's College, Oxford

Some related news.

Background from this journal—

Collegiality, That Hideous Strength , and The Oxford Murders .

See also…

"The heart of the book is the conveying of a meaningful understanding
of where mathematical results originated…."

Friday, December 16, 2011

Take Your Pick

Filed under: General — Tags: , — m759 @ 1:31 PM

Two recent quotes in this journal—

December 14

"Hoban once ruefully observed that death would be a good career move:
'People will say, "Yes, Hoban, he seems an interesting writer, let’s look at him again."'"

December 15

"This poem is concerned with language on a very plain level."

— "Paradoxes and Oxymorons" in Shadow Train

Michael Kinsley in The New York Times  on Sunday, May 13, 2007

Kinsley on the career of Christopher Hitchens—

Interesting! …. Interesting!! …. Interesting!!! …. Interesting!!!!

Where was this train heading?

Kinsley on a book in which Hitchens …

… pronounces the movie “It’s a Wonderful Life” “engaging but abysmal” (a typical Hitchens aside: cleverly paradoxical? witlessly oxymoronic? take your pick)….

Thursday, December 15, 2011

As Is

Filed under: General,Geometry — Tags: , — m759 @ 2:56 AM

What "As" Is —

Image- The Three-Point Line: A Finite Projective Geometry

"This poem is concerned with language on a very plain level."

Shadow Train

"You got to ride it like you find it."
Song lyric

Related entertainment —

IMAGE- Sherlock Holmes: A Game of Shadows

Monday, October 31, 2011

Beauty, Truth, Halloween

Filed under: General — m759 @ 9:00 PM

On Halloween…

"Remember that for Ockham there is nothing in the universe that is
in any way universal except a concept or word: there are no real
natures shared by many things. However, things do resemble one
another, some things more closely  than others. So the various
degrees of resemblance give a foundation in reality for our conceptual
structures, such as Porphyry's tree.
Now resemblance (or similitude or likeness) is a relation.
If such relations are realities, then we can say that there are realities
out there that correspond to our conceptual structures."

R.J. Kilcullen at Macquarie University, course labeled Phil360


"The kernel of a homomorphism is always a congruence.
 Indeed, every congruence arises as a kernel."

Congruence Relation, section on Universal Algebra, in Wikipedia

"Beauty then is a relation."

Gerard Manley Hopkins

"An Attempt to Understand the Problem of Universals"
is the title of a talk by Fabian Geier, University of Bamberg—

"The talk was held at Gdańsk University on May 26th 2008."

Related material— Stevie Nicks turns 60.

Wednesday, October 26, 2011

Erlanger and Galois

Filed under: General,Geometry — m759 @ 8:00 PM

Peter J. Cameron yesterday on Galois—

"He was killed in a duel at the age of 20…. His work languished for another 14 years until Liouville published it in his Journal; soon it was recognised as the foundation stone of modern algebra, a position it has never lost."

Here Cameron is discussing Galois theory, a part of algebra. Galois is known also as the founder* of group theory, a more general subject.

Group theory is an essential part of modern geometry as well as of modern algebra—

"In der Galois'schen Theorie, wie hier, concentrirt sich das Interesse auf Gruppen von Änderungen. Die Objecte, auf welche sich die Änderungen beziehen, sind allerdings verschieden; man hat es dort mit einer endlichen Zahl discreter Elemente, hier mit der unendlichen Zahl von Elementen einer stetigen Mannigfaltigkeit zu thun."

— Felix Christian Klein, Erlanger Programm , 1872

("In the Galois theory, as in ours, the interest centres on groups of transformations. The objects to which the transformations are applied are indeed different; there we have to do with a finite number of discrete elements, here with the infinite number of elements in a continuous manifoldness." (Translated by M.W. Haskell, published in Bull. New York Math. Soc. 2, (1892-1893), 215-249))

Related material from Hermann Weyl, Symmetry , Princeton University Press, 1952 (paperback reprint of 1982, pp. 143-144)—

"A field is perhaps the simplest algebraic structure we can invent. Its elements are numbers…. Space is another example of an entity endowed with a structure. Here the elements are points…. What we learn from our whole discussion and what has indeed become a guiding principle in modern mathematics is this lesson: Whenever you have to do with a structure-endowed entity  Σ try to determine is group of automorphisms , the group of those element-wise transformations which leave all structural relations undisturbed. You can expect to gain a deep insight into the constitution of Σ in this way."

For a simple example of a group acting on a field (of 8 elements) that is also a space (of 8 points), see Generating the Octad Generator and Knight Moves.

* Joseph J. Rotman, An Introduction to the Theory of Groups , 4th ed., Springer, 1994, page 2

Saturday, October 8, 2011

An Ordinary Evening in Hartford

Filed under: General — Tags: , — m759 @ 7:59 AM

From Rebecca Goldstein's Talks and Appearances page—

• "36 (Bad) Arguments for the Existence of God,"
   Annual Meeting of the Freedom from Religion Foundation,
   Marriot, Hartford, CT, Oct 7 [2011], 7 PM

From Wallace Stevens—

"Reality is the beginning not the end,
Naked Alpha, not the hierophant Omega,
of dense investiture, with luminous vassals."

— “An Ordinary Evening in New Haven” VI

For those who prefer greater depth on Yom Kippur, yesterday's cinematic link suggests…

"Yo sé de un laberinto griego que es una línea única, recta."
 —Borges, "La Muerte y la Brújula " ("Death and the Compass")

See also Alpha and Omega (Sept. 18, 2011) and some context from 1931.

Sunday, September 4, 2011

The Found Numbers

Filed under: General — m759 @ 10:01 PM

(A sequel to The Lost Word)

IMAGE-NY Lottery Evening numbers, Sunday, Sept. 4, 2011: 038 and 1919

NY Lottery, Sunday, Sept. 4, 2011

The number "38" might refer, notably, to

  • The old postal code  of Harvard University,
  • the caliber  of one type of handgun…

… and, less notably, to a page number
     that appears in most books.

T.S. Eliot's essay "Tradition and the Individual Talent"
was first published in 1919.*

For some background on that year, see the Harvard Class  of 1919.

For a notable instance of the page number  38, see
Poetry and Belief in the Work of T.S. Eliot ,
by Kristian Smidt (first published in 1949 by
the Norwegian Academy of Science and Letters).

* "Tradition and the Individual Talent" in The Egoist
   at the Modernist Journals Project:
   Part I in Vol. 6, No. 4 (Sept. 1919),
   Parts II-III in Vol. 6, No. 5 (Dec. 1919)

Saturday, August 20, 2011

Castles in the Air

Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

"… the Jews have discovered a way to access a fourth spatial dimension."
— Clifford Pickover, description of his novel Jews in Hyperspace

"If you have built castles in the air, your work need not be lost;
that is where they should be. Now put the foundations under them.”
— Henry David Thoreau

"King Solomon's Mines," 1937—

Image -- The cast of 1937's 'King Solomon's Mines' goes back to the future

The image above is an illustration from  "Romancing the Hyperspace," May 4, 2010.

Happy birthday to the late Salomon Bochner.

Wednesday, July 13, 2011

The Sinatra Code

Filed under: General — Tags: — m759 @ 2:45 AM

From The Da Vinci Code,
by Dan Brown

Chapter 56

Sophie stared at Teabing a long moment and then turned to Langdon.  “The Holy Grail is a person?”

Langdon nodded.  “A woman, in fact.”  From the blank look on Sophie’s face, Langdon could tell they had already lost her.  He recalled having a similar reaction the first time he heard the statement. It was not until he understood the symbology  behind the Grail that the feminine connection became clear.

Teabing apparently had a similar thought.  “Robert, perhaps this is the moment for the symbologist to clarify?”  He went to a nearby end table, found a piece of paper, and laid it in front of Langdon.

Langdon pulled a pen from his pocket.  “Sophie are you familiar with the modern icons for male and female?”  He drew the common male symbol ♂ and female symbol ♀.

“Of course,” she said.

“These,” he said quietly, are not the original symbols for male and female.  Many people incorrectly assume the male symbol is derived from a shield and spear, while the female represents a mirror reflecting beauty.  In fact, the symbols originated as ancient astronomical symbols for the planet-god Mars and the planet-goddess Venus.  The original symbols are far simpler.”  Langdon drew another icon on the paper.




“This symbol is the original icon for male ,” he told her.  “A rudimentary phallus.”

“Quite to the point,” Sophie said.

“As it were,” Teabing added.

Langdon went on.  “This icon is formally known as the blade , and it represents aggression and manhood.  In fact, this exact phallus symbol is still used today on modern military uniforms to denote rank.”

“Indeed.”  Teabing grinned.  “The more penises you have, the higher your rank.  Boys will be boys.”

Langdon winced.  “Moving on, the female symbol, as you might imagine, is the exact opposite.”  He drew another symbol on the page.  “This is called the chalice .”



Sophie glanced up, looking surprised.

Langdon could see she had made the connection.  “The chalice,” he said, “resembles a cup or vessel, and more important, it resembles the shape of a woman’s womb.  This symbol communicates femininity, womanhood, and fertility.”  Langdon looked directly at her now.  “Sophie, legend tells us the Holy Grail is a chalice—a cup.  But the Grail’s description as a chalice  is actually an allegory to protect the true nature of the Holy Grail.  That is to say, the legend uses the chalice as a metaphor  for something far more important.”

“A woman,” Sophie said.

“Exactly.”  Langdon smiled.  “The Grail is literally the ancient symbol for womankind, and the Holy  Grail represents the sacred feminine and the goddess, which of course has now been lost, virtually eliminated by the Church.  The power of the female and her ability to produce life was once very sacred, but it posed a threat to the rise of the predominantly male Church, and so the sacred feminine was demonized and called unclean.  It was man , not God, who created the concept of ‘original sin,’ whereby Eve tasted of the apple and caused the downfall of the human race.  Woman, once the sacred giver of life, was now the enemy.”

“I should add,” Teabing chimed, “that this concept of woman as life-bringer was the foundation of ancient religion.  Childbirth was mystical and powerful.  Sadly, Christian philosophy decided to embezzle the female’s creative power by ignoring biological truth and making man  the Creator.  Genesis tells us that Eve was created from Adam’s rib.  Woman became an offshoot of man.  And a sinful one at that.  Genesis was the beginning of the end for the goddess.”

“The Grail,” Langdon said, “is symbolic of the lost goddess.  When Christianity came along, the old pagan religions did not die easily.  Legends of chivalric quests for the lost Grail were in fact stories of forbidden quests to find the lost sacred feminine. Knights who claimed to be “searching for the chalice” were speaking in codes as a way to protect themselves from a Church that had subjugated women, banished the Goddess, burned nonbelievers, and forbidden pagan reverence for the sacred feminine.”


Happy birthday to Harrison Ford.

One for my baby…



One more for the road.


Sunday, March 27, 2011


Filed under: General — Tags: — m759 @ 2:45 AM

"I just seemed to have more frames per second than other kids."

— Mary Karr, "Facing Altars: Poetry and Prayer"


 See also "Signs and Symbols."

Art based on a cover of Salinger's 'Nine Stories'

Sunday, March 13, 2011


Filed under: General — Tags: — m759 @ 7:59 PM

"Poetry never left me stranded, and as an atheist most of my life, I presumed its mojo was a highbrow, intellectual version of what religion did for those more gullible believers in my midst— dumb bunnies to a one, the faithful seemed to me, till I became one.

In the Texas oil town where I grew up, fierceness won fights, but I was thin-skinned— an unfashionably bookish kid whose brain wattage was sapped by a consuming inner life others didn’t seem to bear the burden of. I just seemed to have more frames per second than other kids."

— "Facing Altars: Poetry and Prayer," by Mary Karr

"The original movie had been slowed to a running time of twenty-four hours.
What he was watching seemed pure film, pure time.
The broad horror of the old gothic movie was subsumed in time."

Point Omega , by Don DeLillo

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